(Page 2 of 6) - Graph Theory: A Topic for Helping Secondary Teachers authored by Wilson, Skip. and Rivera-Marrero, Olgamary.

useful in the real world. Others have stated that discrete mathematics affords students
opportunity to participate actively in the mathematics process and to experience success and
enjoyment in mathematics (Holliday, 1991, Kenny & Bezuszka, 1993). The literature and our
experience also provide evidence that it provides teachers with opportunities to adopt positive
dispositions about innovative teaching strategies (Wilson & Spielman, 2003).

Theoretical Perspective

We believe that innovative teaching involves teachers helping their students to explore

important ideas, solve real problems, make meaningful connections, and work in cooperation
with other students (NCTM, 1989, 2000). Mathematical ideas should be correct because they
make sense and work, not just because a teacher or textbook say so. We believe that student-
centered instruction often requires teachers to share mathematical authority with their students
(Wilson & Lloyd, 2000). Such sharing sometimes involves the use by teachers and their students
of computers and calculators. Rather than simply being told or shown important relationships
and mathematical concepts, with technology students more easily explore and solve meaningful
problems themselves.

Graph theory is a topic that we believe lends itself to such sharing. For example, real-world

applications are extremely prominent in graph theory and the Mathematical Modeling software
(Graubart, C.B., 1997) upon which the example we discuss later in this paper is based,
encourages students to explore the meanings of problems and to represent the problem situations
in multiple ways. The software is dynamic in many of the same ways as Geometer’s Sketchpad
(Jackiw., 1997). For example, when changes in the vertex graph are made, corresponding
changes are displayed in the adjacency/connectivity matrix representation, and vice versa. Partly
because the software enables students to experiment and make sense of problem situations
without having to concentrate on computational aspects, they are more easily able to explore
problems without the teacher’s direct input. Multiple representation and cooperative exploration
in the solving of real-world problems are important components of sharing by teachers of
mathematical authority. Engaging in these processes help students develop sense of
mathematical ideas based on internal voices, not solely on the basis of what their teachers tell
them about conventions and important relationships (King & Kitchener, 1994). To a greater
extent then when simply told by their teachers, students gain an internal voice concerning the
correctness of the concepts considered.

Methods

Many arguments have been presented for including discrete mathematics in the secondary

curriculum. For example, Rosenstein (1997) expressed that discrete mathematics is applicable
(provides different ways to represent real-world problems), accessible (basic mathematics such
as arithmetic is sufficient in order to understand the application of discrete mathematics),
attractive (discrete mathematics problems catch the attention of students and lend themselves to
discovery and exploration) and appropriate (discrete math is for students that are accustomed to
success and for those who are not). Kenny and Bezuszka (1993) stated that discrete mathematics
is appropriate for illustrating and emphasizing NCTM’s process standards. The reader is
encouraged to keep these points in mind as he or she considers the example task described later
in this paper.

A recent study of 15 secondary teachers enrolled in a mathematics course for teachers

(Wilson and Spielman, 2003) suggests how and why graph theory also has positive prospects for
mathematics teacher education. For example, one student said: